Complexity–entropy causality plane analysis of time series generated by a modified nonlinear Langevin equation: Effects of persistence asymmetry, memory length, and nonlinearity

We investigate the ability of the complexity–entropy causality plane (CECP) to discriminate subtle features of temporal structure in time series generated by a modified nonlinear Langevin equation with persistence described by two parameters, p and q . The model provides a unified and flexible stochastic framework in which three key aspects can be varied independently: persistence asymmetry ( 𝑝 − 𝑞 ), memory length ( 𝑚 ), and the degree of nonlinearity of the drift function ( 𝛼 ). For each case, permutation entropy 𝐻 and statistical complexity 𝐶 are computed at embedding dimension 𝑑 𝑥 = 7 and mapped onto the CECP and onto the normalized complexity 𝐶 𝑛 versus 𝐻 plane, the latter eliminating the effect of the entropy-dependent admissible interval. The results show that all three parameters induce systematic, interpretable displacements in the 𝐶 𝑛 – 𝐻 plane, even when the corresponding shifts are small. Increasing persistence asymmetry at fixed 𝑝 raises entropy and lowers 𝐶 𝑛 due to weakened directional bias and enhanced reversals. Increasing memory length 𝑚 reduces both 𝐻 and 𝐶 𝑛 through progressive suppression of admissible ordinal configurations. Decreasing the nonlinearity parameter 𝛼 also reduces both 𝐻 and 𝐶 𝑛 , as a broader drift potential induces long, regular excursions that diminish ordinal diversity. These findings demonstrate that permutation entropy-based analysis is sensitive to subtle stochastic features of time series and can characterize changes in their dynamics that arise from variations in a single model parameter.